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Carleson Measure Estimates and the Dirichlet Problem for Degenerate Elliptic Equations Hofmann, Steve; Le, Phi; Morris, Andrew

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Carleson Measure Estimates and the Dirichlet Problem for Degenerate Elliptic Equations Hofmann, Steve; Le, Phi; Morris, Andrew View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Birmingham Research Portal Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations Hofmann, Steve; Le, Phi; Morris, Andrew License: None: All rights reserved Document Version Peer reviewed version Citation for published version (Harvard): Hofmann, S, Le, P & Morris, A 2018, 'Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations', Analysis and PDE. Link to publication on Research at Birmingham portal Publisher Rights Statement: Checked for eligibility 04/12/2018 This is a peer-reviewed version of an article forthcoming in Analysis and PDE General rights Unless a licence is specified above, all rights (including copyright and moral rights) in this document are retained by the authors and/or the copyright holders. The express permission of the copyright holder must be obtained for any use of this material other than for purposes permitted by law. •Users may freely distribute the URL that is used to identify this publication. •Users may download and/or print one copy of the publication from the University of Birmingham research portal for the purpose of private study or non-commercial research. •User may use extracts from the document in line with the concept of ‘fair dealing’ under the Copyright, Designs and Patents Act 1988 (?) •Users may not further distribute the material nor use it for the purposes of commercial gain. Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document. When citing, please reference the published version. Take down policy While the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has been uploaded in error or has been deemed to be commercially or otherwise sensitive. If you believe that this is the case for this document, please contact [email protected] providing details and we will remove access to the work immediately and investigate. Download date: 01. Feb. 2019 CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM FOR DEGENERATE ELLIPTIC EQUATIONS STEVE HOFMANN, PHI LE, ANDREW J. MORRIS Abstract. We prove that the Dirichlet problem for degenerate elliptic equations n+1 div(Aru) = 0 in the upper half-space (x; t) 2 R+ is solvable when n ≥ 2 and p n the boundary data is in Lµ(R ) for some p < 1. The coefficient matrix A is only assumed to be measurable, real-valued and t-independent with a degener- ate bound and ellipticity controlled by an A2-weight µ. It is not required to be symmetric. The result is achieved by proving a Carleson measure estimate for all bounded solutions in order to deduce that the degenerate elliptic measure is n in A1 with respect to the µ-weighted Lebesgue measure on R . The Carleson measure estimate allows us to avoid applying the method of ϵ-approximability, which simplifies the proof obtained recently in the case of uniformly elliptic co- efficients. The results have natural extensions to Lipschitz domains. Contents 1. Introduction 1 2. Preliminaries 5 3. Estimates for Maximal Operators 11 4. The Carleson Measure Estimate 16 5. Solvability of the Dirichlet Problem 32 5.1. Boundary estimates for solutions 33 5.2. Definition and properties of degenerate elliptic measure 33 5.3. Preliminary estimates for degenerate elliptic measure 35 5.4. The A1-estimate for degenerate elliptic measure 38 5.5. The square function and non-tangential maximal function estimates 43 References 47 1. Introduction We consider the Dirichlet boundary value problem for the degenerate elliptic n+1 equation div(Aru) = 0 in the upper half-space R+ when n ≥ 2 and which we Date: December 3, 2018. 2010 Mathematics Subject Classification. 35J25, 35J70, 42B20, 42B25. Key words and phrases. Square functions, non-tangential maximal functions, harmonic measure, Radon–Nikodym derivative, Carleson measure, divergence form elliptic equations, Dirichlet prob- lem, A2 Muckenhoupt weights, reverse Holder¨ inequality. 1 2 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS make precise below. The boundary Rn × f0g is identified with Rn and we adopt the n+1 n notation X = (x; t) for points X 2 R+ with coordinates x 2 R and t 2 (0; 1). The gradient r := (rx;@t) and divergence div := divx +@t are with respect to all (n + 1)- coordinates. The coefficient A denotes an (n + 1) × (n + 1)-matrix of measurable, n+1 real-valued and t-independent functions on R+ . The matrix A(x):= A(x; t) is not required to be symmetric. We suppose that there exist constants 0 < λ ≤ Λ < 1 n and an A2-weight µ on R such that the degenerate bound and ellipticity (1.1) jhA(x)ξ; ζij ≤ Λµ(x)jξjjζj and hA(x)ξ; ξi ≥ λµ(x)jξj2 hold for all ξ; ζ 2 Rn+1 and almost every x 2 Rn. We use ⟨·; ·⟩ and j · j to denote the n Euclidean inner-product and norm. An A2-weight µ on R refers to a non-negative locally integrable function µ : Rn ! [0; 1] such that ( ˆ )( ˆ ) 1 1 1 µ n = µ < 1; [ ]A2(R ) : sup (x) dx dx Q jQj Q jQj Q µ(x) Rn j j where supQ denotes the supremum over all´ cubes Q in with volume Q . We µ µ = µ also use to denote the measure (Q): ´ Q (x) dx and consider the Lebesgue p n p 1=p space Lµ(R ) with the norm k f k p n := ( j f j dµ) for all p 2 [1; 1). There ffl Lµ(R ) ´ Rn ffl ´ µ = µ −1 µ = j j−1 is also the notation Q f d : (Q) Q f d whilst Q f : Q Q f (x) dx. If µ is identically 1, then A is called uniformly elliptic. The solvability of the Dirichlet problem for general non-symmetric coefficients in that case was obtained only recently by Hofmann, Kenig, Mayboroda and Pipher in [HKMP]. The result in dimension n = 1 had been obtained previously by Kenig, Koch, Pipher and Toro in [KKoPT]. These results assert that for each uniformly elliptic coefficient matrix A, there exists some p < 1 for which the Dirichlet problem is solvable for Lp-boundary data. Conversely, counterexamples in [KKoPT] show that for each p < 1, there exists a uniformly elliptic coefficient matrix A for which the Dirichlet problem is not solvable for Lp-boundary data. In contrast, solvability of the Dirichlet problem for symmetric coefficients in the uniformly elliptic case is well-understood, and we mention only that it was obtained by Jerison and Kenig in [JK] for Lp-boundary data when 2 ≤ p < 1. The solvability of the Dirichlet problem in the uniformly elliptic case has also been established for a variety of complex coefficient structures (see, for instance, [AS, HKMP, HMM]). A significant portion of that theory was recently extended to the degenerate elliptic case by Auscher, Rosen´ and Rule in [ARR] for L2- boundary data. That extension did not include, however, the results for general non-symmetric coefficients in [HKMP]. This paper complements the progress made in [ARR] by extending the solvability obtained for the Dirichlet problem in [HKMP] to the degenerate elliptic case. n+1 n For solvability on the upper half-space R+ , the A2-weight µ on R is extended n+1 µ ; = µ R µ n+1 = µ Rn to the t-independent A2-weight (x t): (x) on (and [ ]A2(R ) [ ]A2( )). We then say that u is a solution of the´ equation div(Aru) = 0 in an open set n+1 1;2 Ω ⊆ R u 2 W Ω n+1 hAru; rΦi = when µ,loc( ) and R+ 0 for all smooth com- Φ 2 1 Ω µ pactly supported functions Cc ( ). The solution space is the local -weighted 1;2 Sobolev space Wµ,loc defined in Section 2. The convergence of solutions to bound- ary data is afforded by estimates for the non-tangential maximal function N∗u of CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 3 solutions u, defined by n (N∗u)(x):= sup ju(y; t)j 8x 2 R ; (y;t)2Γ(x) n+1 where the cone Γ(x):= f(y; t) 2 R+ : jy − xj < tg. If p 2 (1; 1), then the Dirichlet p n problem for Lµ(R )-boundary data, or simply (D)p,µ, is said to be solvable when p for each f 2 Lµ(Rn), there exists a solution u such that 8 n+1 <>div(Aru) = 0 in R+ ; p n (D)p,µ N∗u 2 Lµ(R ); :> limt!0 u(·; t) = f; p where the limit is required to converge in Lµ(Rn)-norm and in the non-tangential n sense whereby limΓ(x)3(y;t)!(x;0) u(y; t) = f (x) for almost every x 2 R . Note that this definition of solvability is distinct from well-posedness, which requires that such solutions are unique. We are able to obtain a uniqueness result for solutions that converge uniformly to 0 at infinity, but the question of well-posedness more generally remains open (see Theorem 5.34 and the preceding discussion). n A non-negative Borel measure ! on a cube Q0 in R is said to be in the A1-class with respect to µ, written ! 2 A1(µ), when there exist constants C; θ > 0, which we call the A1(Q0)-constants, such that ( )θ µ(E) !(E) ≤ C !(Q) µ(Q) for all cubes Q ⊆ Q0 and all Borel sets E ⊆ Q. This is a scale-invariant version of the absolute continuity of ! with respect to µ.
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